Boleslaw Szymanski based on slides by Albert-Lszl Barabsi and - - PowerPoint PPT Presentation

Frontiers of Network Science Fall 2017 Class 19: Robustness II (Chapter 8 in Textbook)

based on slides by Albert-László Barabási and Roberta Sinatra

www.BarabasiLab.com

Boleslaw Szymanski

Attack threshold for arbitrary P(k)

Attack problem: we remove a fraction f of the hubs. At what threshold fc will the network fall apart (no giant component)? Hub removal changes the maximum degree of the network [Kmax  K’max ≤Kmax) the degree distribution [P(k)  P’(k’)] A node with degree k will loose some links because some of its neighbors will vanish. Claim: once we correct for the changes in Kmax and P(k), we are back to the robustness problem. That is, attack is nothing but a robustness of the network with a new K’max and f’. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).

f '= f

2−γ 1−γ

1 ' 1 1 ' − − = κ f

     > > > > > − − =

− −

1 2 2 3 3 3 2

max 2 min 3 max min

γ γ γ γ γ κ

γ γ

K K K K

K'max = Kmin f

1 1−γ

fc

2−γ 1−γ = 2 + 2 −γ

3 −γ Kmin fc

3−γ 1−γ −1

     

c c

f k f k k k − = > < − > < = > < > < = 1 ) 1 ( ' ' '

2 2

κ κ

Network Science: Robustness Cascades

Attack threshold for arbitrary P(k)

Attack problem: we remove a fraction f of the hubs. At what threshold fc will the network fall apart (no giant component)? Cohen et al., Phys. Rev. Lett. 85, 4626 (2000).

Figure: Pastor-Satorras & Vespignani, Evolution and Structure of the Internet: Fig 6.12

fc

• fc depends on γ; it reaches its max for γ<3
• fc depends on Kmin (m in the figure)
• Most important: fc is tiny. Its maximum reaches
• nly 6%, i.e. the removal of 6% of nodes can

destroy the network in an attack mode.

• Internet: γ=2.1, so 4.7% is the threshold.

Network Science: Robustness Cascades

Application: ER random graphs

Consider a random graph with connection probability p such that at least a giant connected component is present in the graph. S surviving giant component Find the critical fraction of removed nodes such that the giant connected component is destroyed. The higher the original average degree, Empty squares show S the larger damage the network can survive. Filled squares l – avg. distance Q: How do you explain the peak in the average distance?

2 c

k 1 1 pN 1 1 1 k k 1 1 f − = − = − − =

S l

Minimum damage

Network Science: Robustness Cascades

1

S

1

f fc Attacks γ ≤ 3 : fc=1

(R. Cohen et al PRL, 2000)

Failures

Albert, Jeong, Barabási, Nature 406 378 (2000)

Summary: Achilles’ Heel of scale-free networks

Network Science: Robustness Cascades

Summary: Achilles’ Heel of complex networks

Internet failure attack

• R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

Network Science: Robustness Cascades

Historical Detour: Paul Baran and Internet

1958

Network Science: Robustness Cascades

A network of n-ary degree of connectivity has n links per node was simulated The simulation revealed that networks where n ≥ 3 had a significant increase in resilience against even as much as 50% node loss. Baran's insight gained from the simulation was that redundancy was the key.

Scale-free networks are more error tolerant, but also more vulnerable to attacks

• squares: random failure
• circles: targeted attack
• S surviving fraction of GC
• l average distance

Failures: little effect on the integrity of the network. Attacks: fast breakdown

S l

Network Science: Robustness Cascades

Real scale-free networks show the same dual behavior

• blue squares: random failure
• red circles: targeted attack
• open symbols: S (size of surviving

component)

• filled symbols: l (average distance)
• break down if 5% of the nodes are eliminated selectively (always

the highest degree node)

• resilient to the random failure of 50% of the nodes.

Similar results have been obtained for metabolic networks and food webs.

S S l l

Network Science: Robustness Cascades

Cascades

• Information cascades

social and economic systems diffusion of innovations

• Cascading failures

infrastructural networks complex organizations

Potentially large events triggered by small initial shocks

Network Science: Robustness Cascades

Cascading Failures in Nature and Technology

Cascades depend on

• Structure of the network
• Properties of the flow
• Properties of the net elements
• Breakdown mechanism

Blackout Flows of physical quantities

• congestions
• instabilities
• Overloads

Earthquake Avalanche

Network Science: Robustness Cascades

Consequences More than 508 generating units at 265 power plants shut down during the

• utage. In the minutes before the

event, the NYISO-managed power system was carrying 28,700 MW of

• load. At the height of the outage, the

load had dropped to 5,716 MW, a loss

• f 80%.

Origin A 3,500 MW power surge (towards Ontario) affected the transmission grid at 4:10:39 p.m.

• EDT. (Aug-14-2003)

Before the blackout After the blackout

Network Science: Robustness Cascades

Northeast Blackout of 2003

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Cascades Size Distribution of Blackouts

Probability of energy unserved during North American blackouts 1984 to 1998.

Source Exponent Quantity North America 2.0 Power Sweden 1.6 Energy Norway 1.7 Power New Zealand 1.6 Energy China 1.8 Energy

Unserved energy/power magnitude (S) distribution

• I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007)

P(S) ~ S −α, 1< α < 2

Network Science: Robustness Cascades

Cascades Size Distribution of Earthquakes

P(S) ~ S −α,α ≈ 1.67

Earthquake size S distribution

• Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003)

Earthquakes during 1977–2000.

Network Science: Robustness Cascades

Failure Propagation Model

• D. Watts, PNAS 99, 5766-5771 (2002)

Initial Setup

• Random graph with N nodes
• Initially each node is functional.

Cascade

• Initiated by the failure of one node.
• fi : fraction of failed neighbors of node i. Node i

fails if fi is greater than a global threshold φ. Erdos-Renyi network

P(S) ~ S −3/2

□ Critical

• Overcritical

Overcritical Undercritical Critical

φ =0.4 f = 0 f = 1/3 f = 1/2 f = 1/2 f = 1/2 f = 2/3

Network Science: Robustness Cascades

<k>

Network falls apart (<k>=1)

Overload Model

Initial Conditions

• N Components (complete graph)
• Each components has random initial load Li

drawn at random uniformly from [Lmin, 1]. Cascade

• Initiated by the failure of one component.
• Component fail when its load exceeds 1.
• When a component fails, a fixed amount P is

transferred to all the rests.

P(S) ~ S −3/2

• I. Dobson, B. A. Carreras, D. E. Newman, Probab. Eng. Inform. Sci. 19, 15-32 (2005)

Critical Overcritical Undercritical Undercritical Overcritical Critical

Li =0.8

P=0.15

Li =0.9 Li =0.7 Li =0.95 Li =1.05 Li =0.85

Network Science: Robustness Cascades

Lmin

Self-organized Criticality (BTW Sandpile Model)

K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)

Homogenous case Scale-free network Initial Setup

• Random graph with N nodes
• Each node i has height hi = 0.

Cascade

• At each time step, a grain is added at a randomly

chosen node i: hi ← hi +1

• If the height at the node i reaches a prescribed

threshold zi = ki, then it becomes unstable and all the grains at the node topple to its adjacent nodes: hi = 0 and hj ← hj +1

• if i and j are connected.

Homogenous network: <k2> converges

P(S) ~ S −3/2

Scale-free network : pk ~ k-γ (2<γ<3)

P(S) ~ S −γ/(γ −1)

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Branching Process Model

Branching Process

Starting from a initial node, each node in generation t produces k number of

• ffspring nodes in the next t + 1

generation, where k is selected randomly from a fixed probability distribution qk=pk-1.

Hypothesis

• No loops (tree structure)
• No correlation between branches

K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)

Narrow distribution: <k2> converged

P(S) ~ S −3/2

Fat tailed distribution: qk ~ k-γ (2<γ<3)

P(S) ~ S −γ/(γ −1)

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Fix <k>=1 to be critical  power law P(S)

Short Summary of Models: Universality

Models Networks Exponents Failure Propagation Model ER 1.5 Overload Model Complete Graph 1.5 BTW Sandpile Model ER/SF 1.5 (ER) γ/(γ - 1)(SF) Branching Process Model ER/SF 1.5 (ER) γ/(γ - 1)(SF)

P(S) ~ S −3/2

Universal for homogenous networks Same exponent for percolation too (random failure, attacking, etc.)

Network Science: Robustness Cascades

Explanation of the 3/2 Universality

Simplest Case: q0 = q2 = 1/2, <k> = 1

S: number of nodes X: number of open branches k= 0 k=2 S = S+1 X = X -1 S = S+1 X = X+1 ½ chance ½ chance S = 1 X = 1

S = 2, X = 0 S = 2, X = 2

X >0, Branching process stops when X = 0 X S Dead

Explanation of 3/2 Universality

Question: what is the probability that X = 0 after S steps? First return probability ~ S-3/2

• M. Ding, W. Yang, Phys. Rev. E. 52, 207-213 (1995)

Equivalent to 1D random walk model, where X and S are the position and time , respectively. X S Dead

Size Distribution of Branching Process (Cavity Method)

S1 S2 S1 S = 1+S1 S = 1+S1+S2 S = 1 k = 0 k = 1 k = 2 K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physica A 346, 93-103 (2005)

 + − + + + − + +

∑ ∑

2 1 1

, 2 1 2 1 2 1 1 1

) 1 ( ) ( ) ( ) 1 ( ) ( ) 1 (

S S S

S S S S P S P q S S S P q q δ δ δ = ) (S P

Network Science: Robustness Cascades

Solving the Equation by Generating Function

Phase Transition <S> = GS’(1) = 1+ Gk’(1) GS’(1) = 1 + <k> <S>, then <S> = 1/(1- <k>) The average size <S> diverges at <k>c = 1 Definition:

GS(x) = ΣS=0 P(S)xS Gk(x) = Σk=0 qkxk

Property:

GS(1) = Gk(1) = 1 GS’(1) = <S>, Gk’(1) = <k>

∑ ∑ ∑

        − + =

= k S S k j j k k

k

S S S P S P S P q S P





, 1 2 1

1

) 1 ( ) ( ) ( ) ( ) ( δ

∑ ∑ ∑

=         ∑ =

+ k k S k k S S S k k S

x xG q x S P S P q x G

k j j

) ( ) ( ) ( ) (

, 1 1

1 

 )) ( ( x G xG

S k

=

Theorem:

If P(k) ~ k-γ (2<γ<3), then for δx < 0, |δx| << 1 G(1+ δx) = 1 + <k>δx + <k(k-1)/2> (δx)2 + … + O(|δx|γ - 1)

Finding the Critical Exponent from Expansion

P(S) ~ S −α,1< α < 2 GS(1+ δx) ≈ 1 + A|δx|α -1

Homogenous case: <k2> converged <k> = 1, <k2> < ∞

Gk(1+ δx) ≈ 1 + δx + Bδx2

Inhomogeneous case: <k2> diverged <k> = 1, qk ~ k-γ (2<γ<3)

Gk(1+ δx) ≈ 1 + δx + B|δx|γ - 1

Definition:

GS(x) = ΣS=0 P(S)xS Gk(x) = Σk=0 qkxk

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Critical Exponent for Homogenous Case

GS(x) = xGk(GS(x))

Homogenous case

Gk(1+ δx) ≈ 1 + δx + Bδx2 GS(1+ δx) ≈ 1 + A|δx|α -1 GS(x) ≈ 1 + A|δx|α -1 xGk(GS(x)) ≈ (1+δx)[1+ (GS(1+δx)-1) + B (GS(1+δx)-1)2] ≈ (1+δx)[1+ A|δx|α -1 + AB|δx|2α -2] = 1 + A|δx|α -1 + AB|δx|2α -2 + δx + O(|δx|α) The lowest order reads AB|δx|2α -2 + δx = 0, which requires 2α -2 = 1and A = 1/B. Or, α = 3/2

Network Science: Robustness Cascades

Critical Exponent for Inhomogeneous Case

GS(x) = xGk(GS(x))

Inhomogeneous case

Gk(1+ δx) ≈ 1 + δx + B|δx|γ - 1 GS(1+ δx) ≈ 1 + A|δx|α -1 GS(x) ≈ 1 + A|δx|α -1 xGk(GS(x)) ≈ (1+δx)[1+ (GS(1+δx)-1) + B |GS(1+δx)-1|γ -1] ≈ (1+δx)[1+ A|δx|α -1 + AB|δx|(α -1)(γ -1)] = 1 + A|δx|α -1 + AB|δx|(α -1)(γ -1) + δx + O(|δx|α) The lowest order reads AB|δx|(α -1)(γ -1) + δx = 0, which requires (α -1)(γ -1) = 1and A = 1/B. Or, α = γ/(γ −1)

Network Science: Robustness Cascades

Compare the Prediction with the Real Data

Source Exponent Quantity North America 2.0 Power Sweden 1.6 Energy Norway 1.7 Power New Zealand 1.6 Energy China 1.8 Energy

Blackout

• I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007)

Earthquake α ≈ 1.67

• Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003)

   < < − > =

−

3 2 ), 1 /( 3 , 2 / 3 , ~ ) ( γ γ γ γ α

α

S S P

Blackout

Network Science: Robustness Cascades